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Independent and Dependent Events

Concept

Independent Events

Two events A and B are independent events if the occurrence of one event does not affect the occurrence of the other. It is also said that they are independent if and only if the probability that both events occur is equal to the product of the individual probabilities.

P(A ⋂ B)=P(A)* P(B)

Why

For example, suppose a bowl contains three marbles, one green, one orange, and one blue. Someone wants to find the probability of first drawing a green marble, then an orange one. The marbles will be drawn one at a time.

Let G, B, and O be the events of drawing green, blue, and orange marbles, respectively. The probability of first picking a green marble can be calculated by dividing the favorable outcomes by the possible outcomes. The bowl currently contains 3 marbles in total, 1 of which is green. P( G)= 1/3 Suppose that the first marble is replaced before the second draw. After the replacement of the first marble drawn, there are again 3 marbles in the bowl, 1 of which is orange. P( O)= 1/3 Note that there are 9 possible outcomes for drawing two marbles if the marbles are drawn one at a time and then replaced before the next drawing. Only 1 of these options corresponds to an event of drawing a green marble and then an orange marble. G G G B G O B B B G B O O O O G O B Therefore, the combined probability of picking a green marble first and an orange marble second is 19. Since the probability of both events occurring equals the product of the individual probabilities, the events are independent. 1/3* 1/3 = 1/9 [0.3em] ⇓ [0.3em] P( G)* P( O)=P( G⋂ O)

Concept

Dependent Events

Two events A and B are considered dependent events if the occurrence of either event affects the occurrence of the other. If the events are dependent, the probability that both events occur is equal to the product of the probability of the first event occurring and the probability of the second event occurring after the first event.

P(A ⋂ B)=P(A)* P(B|A)

Why

For example, suppose a bowl contains three marbles, one green, one orange, and one blue. Someone wants to find the probability of first drawing the green marble, then the orange one. The marbles will be drawn one at a time.

Let G, B, and O be the events of drawing green, blue, and orange marbles, respectively. The probability of first picking the green marble can be calculated by dividing the favorable outcomes by the possible outcomes. The bowl currently contains 3 marbles in total, 1 of which is green. P( G)= 1/3 Suppose that after the green marble is drawn, it is not replaced in the bowl.

This affects the probability of picking the orange marble on the second draw. Now there is still 1 orange marble in the bowl, but instead of 3, there are 2 marbles in total in the bowl. P( O| G)= 1/2 The sample space of the situation can be found using this information. G B G O B G B O O G O B Out of the 6 possible outcomes, only 1 outcome corresponds to first drawing the green marble and then the orange marble. Therefore, the probability of picking the green and then the orange marble is 16. 1/3* 1/2 = 1/6 [0.3em] ⇓ [0.3em] P( G) * P( O| G) = P( G ⋂ O) Because the occurrence of the first event affects the occurrence of the second, these events can be concluded to be dependent.

Method

Determining if the Events Are Independent

To determine if two events A and B are independent, it should be checked whether they satisfy the following rule. P(A andB)=P(A)* P(B) For example, consider the probability that a newborn baby is a girl and is born on a Wednesday. To conclude that the events are independent, there are four steps to follow.

1

Calculate the Probability of Both Events

To calculate the probability that both events occur, the favorable outcomes are divided by the possible outcomes. First, the possible outcomes for a baby's biological gender is boy or girl.

Second, the possible outcomes to be born on a specific day in a week is 7, since there are 7 different days in a week.

Therefore, there is a total of 14 possible outcomes and only one favorable outcome that the baby is a girl and is born on a Wednesday.

The probability that a baby is a girl and is born on a Wednesday day is the quotient of the numbers of favorable and possible outcomes. P(G and W)= 114

2

Calculate the Probability of Each Event

Next, the probability of each individual event can be calculated. The probability that a newborn baby is a girl is 1 out of 2. P(G)=1/2 Since there are 7 days in a week and only 1 day is Wednesday, the probability to be born on a Wednesday is 1 out of 7. P(W)=1/7

3

Find the Products of the Probabilities

The next step is to find the product of probabilities for each event. To do so, multiply the individual probabilities from the previous step. P(G)* P(W)=1/2*1/7=1/14

4

Compare

The probability that both events occurs P(G and W) is 114 and the product of the two separate events is also 114. P(G and W)=P(G)* P(W)=1/14 Therefore, the Multiplication Rule of Probability is satisfied, which means that the events are independent.

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